Читать книгу Hydraulic Fluid Power - Andrea Vacca - Страница 69

Another important application of the momentum equation is for the determination of the so‐called flow forces in hydraulic control valves.

The flow forces act on the moving element (generally a spool or a poppet) of the valve, and they are generated by the flow through the component. As it will be mentioned in Chapter 8, the presence of flow forces can significantly affect the operation of hydraulic control valves as well as the design of the valve actuation mechanism.

To understand the nature of flow forces and derive an analytical expression that can be useful to study most of the typical geometries of hydraulic control valves, the simplified case in Figure 3.17 is taken as reference. The figure represents a spool that varies the valve opening between the pump (1) supply and the work port (2). Specifically, the spool can move horizontally along the x axis, and in the position represented in the figure, it determines an effective opening area between the valve ports respectively at the sections indicated in figure with 1 (inflow) and 2 (outflow).

The figure considers the common situation of A₁ ≫ A₂ so that, per the conservation of mass, higher exit velocities are obtained. This is a very typical working condition of many hydraulic control valves, when low flow areas are implemented to regulate the actuator speed. The figure clearly shows how the flow changes its velocity and direction (therefore, its momentum) as it crosses the valve. Most of this momentum change is caused by the spool itself. The important question to be answered is: what is the force acting on the spool because of this momentum change? The answer is very important to understand how to size the valve actuation system, namely, the system that sets the position of the spool, which can be manually operated, electrically operated, or hydraulically operated. More details on the valve actuation systems will be provided in Chapter 8.

Figure 3.17 Reference geometry for the analysis of flow forces.

One possible way to answer this question consists in solving the pressure distribution at the spool walls, which is qualitatively shown in Figure 3.17. This approach requires a proper differential flow approach of analysis, where the governing equations are written for a differential fluid element and numerically integrated by mean of a computational fluid dynamics (CFD) tools. Numerical CFD techniques also allows studying more complex geometries that sometimes occurs in modern hydraulic control valves. However, the numerical CFD analysis can be time consuming, and it does not provide an analytical expression of the flow force. This analytical expression can be very useful to the valve designer to gain an intuitive understanding of the development of the flow forces, and more importantly it can be used if to formulate the proper controller parameters in case of hydraulic control valves using closed‐loop controls.

An analytical expression can be easily calculated applying the momentum equation to a properly selected CV. By considering the annular shaped CV indicated in Figure 3.17, with constant fluid density, the equation becomes

(3.44)

Assuming uniform flow at both inlet and outlet sections,

(3.45)

where and are the unit vectors, respectively, along the horizontal and vertical directions. For a rigorous application of the momentum Eq. (3.43), the CV extends outward from the spool exit section A₂, until the vena contraction, Ω. This section is defined when the flow is mono‐dimensional (1D), which means parallel velocity streamlines. Only at Ω the uniform flow assumption is valid to describe the flow exiting the CV.

Therefore, from Eq. (3.44),

(3.46)

The objective of the flow force analysis usually corresponds to the force acting along the direction of motion of the moving element (in this case, the horizontal axis). This is the force that the actuation mechanism has to counter‐react to balance the spool. The vertical force is usually compensated through symmetric design of the spool and housing. For example, in this case, a second exit area would be typically present in the lower end of the spool. Therefore, for the case in Figure 3.17, the attention is focused on the horizontal component of Eq. (3.46):

(3.47)

The term F_x corresponds to the forces acting on the CV through the CS. These include the pressure forces on the inlet and outlet sections, the pressure forces at the walls, and the frictional forces due to fluid shear. The pressure forces at both sections 1 and 2 can be considered as vertical³, and typically the effects of the fluid shear inside the valve is negligible when compared to the flow forces. For this reason, the above expression (3.47) summarizes all the forces that the walls exert on the fluid inside the CV.

The flow force F_fl can be seen as the reaction force of the above F_x, which is the force that the fluid exerts to the bounding surfaces of the CV.

(3.48)

The flow force presents both a stationary component and a transient component. The stationary component corresponds to a given position of the spool or the poppet of the valve and a constant flow rate. The transient component relates to variations of the spool (or poppet) position, as well as flow variations. In typical problems, the transient component is neglected.

The first term refers to transient conditions, and it can be neglected when studying a stationary condition. Therefore, when studying hydraulic control valves, the second term, is normally the most important one.

(3.49)

The negative sign in Eq. (3.48) implies that for the spool valve geometry of Figure 3.17, the flow force tends to reduce the opening area of the spool toward the outlet section 2. This is also qualitatively visualized in the details of Figure 3.17, which shows the two different pressure distributions at the two opposite spool lands: a uniform distribution at the left side, which comes from low fluid velocities, and a gradually decreasing pressure distribution at near the exit land, which results from an accelerating flow.

According to Eq. (3.48), the amount of the flow force is linked to the flow rate with a quadratic relation, meaning that flow forces can become severe at high flow rates. Additionally, the flow force depends on the exit angle θ of the flow, often referred as flow jet angle, which is usually a quantity difficult to estimate. Luckily, multiple studies on the flow jet angle for different valve geometries are available in the literature, with Merritt's textbook being a meaningful example [32]. According to this source, the angle to be used for geometries such as the one in Figure 3.17 is 69°. This value, however, can be slightly affected by the amount of opening and by the clearance between the spool and the valve body.

The following example shows how the evaluation of the direction of the flow force might not be straightforward, even for very simple geometric cases.